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Lösung 3.2:3

Aus Online Mathematik Brückenkurs 2

(Unterschied zwischen Versionen)
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K (Lösning 3.2:3 moved to Solution 3.2:3: Robot: moved page)
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{{NAVCONTENT_START}}
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If we mark the three complex numbers in the plane, we see that the fourth corner will have
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<center> [[Image:3_2_3-1(2).gif]] </center>
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<math>\text{3}+\text{2}i</math>
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{{NAVCONTENT_STOP}}
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and
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{{NAVCONTENT_START}}
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<math>\text{3}i</math>
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<center> [[Image:3_2_3-2(2).gif]] </center>
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as neighbouring corners.
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{{NAVCONTENT_STOP}}
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[[Image:3_2_3_1.gif|center]]
[[Image:3_2_3_1.gif|center]]
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In order to find the fourth corner, we use the fact that in a square opposite sides are parallel and all sides have the same length. This means that the vector from
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<math>\text{1}+i</math>
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to
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<math>\text{3}i</math>
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is equal to the vector from
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<math>\text{3}+\text{2}i</math>
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to the fourth corner.
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[[Image:3_2_3_2.gif|center]]
[[Image:3_2_3_2.gif|center]]
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If we interpret the complex numbers as vectors, this means that the vector from
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<math>\text{1}+i</math>
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to
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<math>\text{3}i</math>
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is
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<math>3i-\left( 1+i \right)=-1+2i</math>
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And we obtain the fourth corner if we add this vector to the corner
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<math>\text{3}+\text{2}i</math>,
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<math>\text{3}+\text{2}i+\left( -1+2i \right)=2+4i</math>

Version vom 15:12, 22. Okt. 2008

If we mark the three complex numbers in the plane, we see that the fourth corner will have 3+2i and 3i as neighbouring corners.


In order to find the fourth corner, we use the fact that in a square opposite sides are parallel and all sides have the same length. This means that the vector from 1+i to 3i is equal to the vector from 3+2i to the fourth corner.


If we interpret the complex numbers as vectors, this means that the vector from 1+i to 3i is


3i−1+i=−1+2i 


And we obtain the fourth corner if we add this vector to the corner 3+2i,


3+2i+−1+2i=2+4i 

Von „http://wiki.math.se/wikis/2009/bridgecourse2-TU-Berlin/index.php/L%C3%B6sung_3.2:3“